Some completely monotonic functions relating to the gamma function

Abstract

We propose a new sharp family of approximations which improve the results of Sandor and Debnath [On certain inequalities involving the constant e and their applications, J. Math. Anal. Appl. 249 (2000), pp. 569–582], Batir [Sharp inequalities for factorial n, Proyecciones 27 (2008), pp. 97–102] and other recent results. Then, we prove the complete monotonicity of a function involving the gamma function to establish new sharp upper and lower bounds for the gamma and the digamma functions.